{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EAWVHG6JFT2KVUV2QJJAKGYOZM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"39889368ab56e664fb1ee7a16e9aa1b044424515b40c4b5e7a85d48156434bf6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-09-17T16:33:55Z","title_canon_sha256":"9a14807f4812963cd46ae621b0697fa888ab10bc5a7f0e1c9b8a1eb83d2d6db6"},"schema_version":"1.0","source":{"id":"1309.4375","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4375","created_at":"2026-05-18T03:13:08Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4375v1","created_at":"2026-05-18T03:13:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4375","created_at":"2026-05-18T03:13:08Z"},{"alias_kind":"pith_short_12","alias_value":"EAWVHG6JFT2K","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EAWVHG6JFT2KVUV2","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EAWVHG6J","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:9003b9e16cf2f577e857ab5e573f50ef42743ba60c79508f2774b458ed92c043","target":"graph","created_at":"2026-05-18T03:13:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce a notion of joint spectrum for a tuple of compact operators on a separable Hilbert space and show that in many situations these operators commute if and only if the joint spectrum consists of countably many, locally finite, complex hyperplanes. In particular, we show that normal matrices (of the same size) $A_1,\\cdots,A_n$ commute if and only if the polynomial $\\det(z_1A_1+\\cdots+z_nA_n+I)$ is completely reducible, that is, it can be factored into a product of linear polynomials.","authors_text":"Isaak Chagouel, Kehe Zhu, Michael Stessin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-09-17T16:33:55Z","title":"Geometric spectral theory for compact operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4375","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5cab990393b67d43876bf0bbce22027d5118ded39ef1585dbd4bcfbb3957c306","target":"record","created_at":"2026-05-18T03:13:08Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"39889368ab56e664fb1ee7a16e9aa1b044424515b40c4b5e7a85d48156434bf6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-09-17T16:33:55Z","title_canon_sha256":"9a14807f4812963cd46ae621b0697fa888ab10bc5a7f0e1c9b8a1eb83d2d6db6"},"schema_version":"1.0","source":{"id":"1309.4375","kind":"arxiv","version":1}},"canonical_sha256":"202d539bc92cf4aad2ba8252051b0ecb0b9ffb5e771cf529315acee4d125e5bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"202d539bc92cf4aad2ba8252051b0ecb0b9ffb5e771cf529315acee4d125e5bf","first_computed_at":"2026-05-18T03:13:08.913052Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:08.913052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"chHAYz60yrnNKOar4s3TcuorA0ZpUCcB9q327ma4XpOi09OtbuSarGmH3YCRXhoNBfrAmlLcArFaPXAn47XLAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:08.913871Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.4375","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5cab990393b67d43876bf0bbce22027d5118ded39ef1585dbd4bcfbb3957c306","sha256:9003b9e16cf2f577e857ab5e573f50ef42743ba60c79508f2774b458ed92c043"],"state_sha256":"a22fdb04cae8a4bc0196f3e14f94f60c3287005dbde93106f139a52dfbc0e9d2"}